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2 Contents 1 Context 4 2 Density of Hyperbolicity Connection with the closing lemma Quasi-conformal rigidity How to prove rigidity? The strategy of the proof of QC-rigidity Maps which are typical in the sense of Lebesgue Maps with a randomly chosen parameter are not hyperbolic The really bad maps Part II of the Palis conjecture. Is it typical for physical measures to exist? Existence of ACIM Strategy of proof of the theorem on the existence of acim s without growth conditions

3 4.2 The proof of Proposition Ubiquity of having a physical measure Strategy of proof Reluctant versus persistent recurrence The Decaying Geometry Condition Combinatorial classification Parameters in DG Robustness under changes in the parameter Decay of Geometry and the SC condition Going from Dynamical to Parameter Space Parameter removal

4 1 Context We will consider here smooth dynamical systems f : M M on some smooth compact manifold M of finite dimension. One of the central aims in dynamical systems is to provide a classification, and perhaps even to show that typical systems don t have pathological behaviour. One obvious classification is the following: 1. f is hyperbolic or Axiom A. 2. f is not hyperbolic, but f has a physical measure. Roughly speaking, this means that for typical starting points x and for continuous observables φ, the sequence 1 n 1 n i=0 φ(f i (x)) converges. 3. other, bad maps. The most ambitious hope would to show that hyperbolic maps are dense. Apparently, up to the late 1960 s, Smale believed that hyperbolic systems are dense in all dimensions, but this was shown to be false in the early 1970 s for diffeomorphisms on manifolds of dimension 2 (by Newhouse and others). Whether the space of non-bad 4

5 maps (in the above sense) is dense is still not known; not even in the C 1 topology. If dim(m) = 1 much more is known. This is the topic of these talks. 2 Density of Hyperbolicity The problem of density of hyperbolicity in dimension one goes back in some form to Fatou (in the 1920 s). Smale gave this problem naively as a thesis problem in the 1960 s (Guckenheimer s thesis). The problem whether hyperbolicity is dense in dimension one was studied by many people, and it was solved in the C 1 topology by Jakobson, a partial solution was given in the C 2 topology by Blokh+Misiurewicz and C 2 density was finally proved by Shen in 2003 or so. For quadratic maps f a = ax(1 x), the Fatou conjecture states that the periodic windows are dense in the following diagram (where horizontally a is drawn, and vertically [0, 1] and iterates iterates of a typical starting point. 5

6 Theorem 1. Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real one-dimenional map f is hyperbolic if each critical point is in the basin of a hyperbolic periodic point. This implies that the real line is the union of a repelling hyperbolic set (a Cantor set of zero Lebesgue measure), the basin of hyperbolic attracting periodic points and the basin of infinity. So for a.e. point x, the iterates f n (x) tend to a periodic orbit. 6

7 In the quadratic case the above theorem says that for a dense set of real parameters a, the non-wandering set of x ax(1 x) is hyperbolic. This quadratic case was proved during the mid 90 s by Lyubich, Graczyk and Swiatek. But this case is special, because in this case certain return maps become almost linear. This special behaviour does not even hold for maps of the form x x 4 + c. Note that the above theorem implies that the space of hyperbolic polynomials is an open dense subset in the space of real polynomials of fixed degree. Every hyperbolic map satisfying the mild no-cycle condition (critical points are not eventually mapped onto other critical points, are structurally stable). The above theorem allows us to solve the 2nd part of Smale s eleventh problem for the 21st century. Theorem 2. Hyperbolic maps are dense in the space of C k maps of the compact interval or the circle, k = 1, 2,...,, ω. Theorem 3. Any complex polynomial which is not infinitely often renormalizable, can be approximates by a hyperbolic polonomial. 7

8 (If we could prove this without the condition that the map is only finitely renormalizable, the complex Fatou conjecture would follows.) 2.1 Connection with the closing lemma Theorem 4 (Pugh s Closing Lemma). Let x be a recurrent point of a smooth diffeomorphism f on a compact manifold. Then there exists a smooth diffeomorphism g which is C 1 close to f for which x is periodic. For the last 30 years any attempt to prove the C k version of this result has been unsuccessful. However, one of the consequences of our result is the one-dimensional version: Theorem 5 (1-D Closing Lemma). Assume k = 1, 2,...,, ω and let x be a recurrent point of a C k interval map f. Then there exists a smooth map g which is C k close to f for which x is periodic. 8

9 2.2 Quasi-conformal rigidity The proof of these result heavily depends on complex analysis. In fact the theorems above can be derived from the following rigidity result. Theorem 6. Let f and f be real polynomials of degree n which only have real critical points. If f and f are topologically conjugate (as dynamical systems acting on the real line) and corresponding critical points have the same order, then they are quasiconformally conjugate (on the complex plane). (A critical point c is a point so that f (c) = 0. Not all critical points of a real polynomial need to be real.) If the polynomials are not real, then we need to make an additional assumption: Theorem 7. Let f and f be complex polynomials of degree n which are not infinitely renormalizable and only have hyperbolic periodic points. If f and f are topologically conjugate, then they are quasiconformally conjugate. 9

10 (This generalizes the famous theorem of Yoccoz, proving that the Mandelbrot set associated to the quadratic family z z 2 + c is locally connected at non-renormalizable parameters.) It would be nice to say that the polynomials are C 1 conjugate: but this is obviously wrong: it would imply that eigenvalues of corresponding periodic points are the same. What might be true is the following: Conjecture: Assume that f, f : R R are C and conjugate with corresponding critical points of the same (finite) order. If ω is a minimal set for f (i.e., all orbits inside ω are dense in ω), then there exists a C 1 homeomorphism h: R R such that h f = f h restricted to ω. The best thing one can hope for is (global) quasiconformally conjugacy: This means that there exist a conjugacy h: C C between f, f : C C and a constant K < such that for Lebesgue almost all x C lim inf r 0 sup y x =r h(y) h(x) inf y x =r h(y) h(x) < K. 10

11 y x h h(x) h(y) Such maps are, for example, Hölder and in Lebesgue almost every point they are differentiable (as maps from C = R 2 to C = R 2 ). 2.3 How to prove rigidity? As usual, we shall try to create a Markov map. 11

12 In the weakly hyperbolic setting one can hope for a countable Bernoulli description: a partition I i of the relevant dynamical space I I = I i (disjoint union) so that f n i : I i I has bounded distortion. In some cases we can, but in many cases this is hopeless: the only way to get f n i : I i I surjective would be to allow not only for unbounded distortion, but even for unbounded degree. The next best thing is to have a countable Markov description: to construct a partition I i so that f n i I i maps onto a union of partition elements I j. One can view this as a random walk. Here, and in general in holomorphic dynamics, we use essentially this approach: We take a sequence of partitions P n of the complex plane with the property that some iterate of f maps elements of the partition P n onto a union of elements of P n 1. 12

13 13

14 The boundaries of the puzzles consist of special curves and there is a natural parametrization on their boundary. (Any polynomial f of degree d is conformally conjugate to z z d near. W.r.t. these coordinates the rays correspond to straight lines, and the equipotentials to circles. In particular, f and f are conformally conjugate near 14

15 and this conjugacy is natural w.r.t. the rays and equipotentials.) The main technical hurdle in our paper is to obtain control on the shape of the puzzle-pieces of this Markov map. That is enough, because of a new way in which we are able to construct quasiconformal conjugacies: Theorem 8 (QC-Criterion). For any constant ɛ > 0 there exists a constant K with the following properties. Let φ: Ω Ω be a homeomorphism between two Jordan domains. Let X be a subset of Ω consisting of pairwise disjoint topological open discs X i. Assume moreover, 1. for each i both X i and φ(x i ) have ɛ-bounded geometry and moreover 2. φ is conformal on Ω X i. mod(ω X i ), mod( Ω φ(x i )) ɛ Then there exists a K-qc map ψ : Ω Ω which agrees with Ω on the boundary of Ω. 15

16 2.4 The strategy of the proof of QC-rigidity So the proof of the rigidity theorem relies on the following steps: 1. Associate to the polynomial f a suitable sequence of partitions P n. 2. Let Ω n be a union of puzzle piece containing the critical points, and show that one has control of the domains of the first return map to Ω n, as in the previous criterion. This is only true provided one constructs the sequence of partitions P n very carefully. 3. Do the same for the topologically conjugate polynomial f. 4. Because of the above properties and the QC-criterion there exists a qc homeomorphism h n : Ω n Ω n which preserves the natural parametrization on the boundary; 5. Because h n : Ω n Ω n is natural on the boundary, the above qc map h n can be extended to a global homeomorphism h n 16

17 which is quasi-conjugacy and so that for each x / Ω n. h n f(x) = f h n (x) 6. Since K-qc homeomorphisms form a compact space, we can extract a K-qc limit of the sequence h n. As Ω n shrinks to a point, the limit h is a conjugacy. By far the hardest part is to prove moduli estimates. Our original proofs can now be simplified using a lemma of Lyubich and Kahn, and can be applied to complex maps which are not infinitely renormalizable. 17

18 3 Maps which are typical in the sense of Lebesgue 3.1 Maps with a randomly chosen parameter are not hyperbolic The set of hyperbolic polynomials does not have full measure with the space of all polynomials. This is a consequence of a theorem of Jakobson from the early 1980 s. This theorem states that Theorem 9. The set of parameters a for which f a (x) = ax(1 x) has an absolutely continuous invariant (probability) measure µ has positive Lebesgue measure. So for a randomly chosen parameter a the orbits of f a (x) = ax(1 x) do NOT go to a periodic orbit. If there exists the invariant measure as in the theorem, one still can make statistical statements: For example the frequency of times iterates visit some open interval U is equal to µ(u). Even better, for any smooth function φ: R R 18

19 one has lim n 1 n n 1 k=0 φ(f k (x)) = φ dµ for almost all x. So in this case, even if one cannot forecast what happens long ahead of time, one can give predictions about averages. 3.2 The really bad maps In the one-dimensional case, the Palis conjecture states that most maps are not bad : they form a set of zero Lebesgue measure in the space of all polynomials. Of course there are many bad maps. Hofbauer and Keller gave several examples of quadratic maps f which have various types of bad behaviour: for some bad maps, typical orbits stay most of the time near 1 a repelling orbit: for a.e. x, n 1 n i=0 δ f i (x) converges to a repelling orbit; 19

20 1 for some other bad maps, for a.e. x, n 1 n i=0 δ f i (x) does not converge in any sense. This sort of bahaviour is bad. It means that no sort of long-term prediction is possible. Not even about averages. 3.3 Part II of the Palis conjecture. Is it typical for physical measures to exist? One of the main challenges in the theory of dynamical systems is to solve the following: Question: Is it true that C k -generically, a diffeomorphism has at least one physical measure (and at most finitely many)? Here we say that an f-invariant measure µ is physical or SRB, if the set B(µ) of points x such that for every continuous functions φ: R R one has lim n 1 n n 1 k=0 φ(f k (x)) = 20 φdµ

21 has positive Lebesgue measure. Definite progress was made towards this question if we are allowed to make C 1 perturbations, by a number of people. However in the C k case, k 2, no progress seems in sight. In the one-dimensional case there is much more progress as will see. 21

22 4 Existence of ACIM Proving the existence of invariant goes back a long way. The hyperbolic case. One of the classical cases one considers is when there is a partition I i of an interval I I = I i (disjoint union) so that g : I i I is a C 2 diffeomorphism which is expanding g (x) > λ > 1. More precisely one asks log Dg(x) log Dg(y) K f(x) f(y). Combining this with the the expansion gives that all iterates of g have bounded distortion: there exists K 1 so that (BDI) log Dg n (x) log Dg n (y) K 1 for x, y in some domain of continuity of g n. For proving that there is an absolutely continuous invariant measure it is enough to have that for each ɛ > 0 there exists δ > 0 so that if A is a Borel set with λ(a) < ɛ then λ(f n (A)) < δ for all n 0. This follows 22

23 immediately from (BDI). In fact one can prove more: there exists the density has bounded variation. So one can define a Perron-Frobenius operator P : E E on a suitable Banach space of functions (i.e. densities of measures) in such a way that P (τ) is the density of the push-forward g (τ dλ) of the measure τ dλ. This operator will have an eigenvalue equal to one (corresponding to the density of an invariant measure) and - if all is well - the rest of the spectrum is well-inside the unit circle. Difficulties in the non-hyperbolic case. For more general maps there are three problems: 1. images of branches can become short 2. the critical points means that if there is a density for an invariant measure, it will definitely have poles; so what space to work in? 3. even the map has unbounded distortion; so to control the nonlinearity of iterates of branches is even more difficult. 23

24 Some prior results. Misiurewicz proved that an acip exists for an S-multimodal map without periodic attractors or recurrent critical points. Collet-Eckman proved that an S-unimodal map f satisfying the following condition (the Collet-Eckmann condition) has acip: (CE) lim inf n log (f n ) (f(c)) n > 0, where c denotes the critical point of f. Nowicki and I improved this by showing that the following summability condition guarantees the existence of an acip for an S-unimodal map: (NS) n=0 1 <, (f n ) (f(c)) 1/l where l is the order of the critical point c. Moreover, it was proved that the density of the acip with respect to the Lebesgue measure belongs to L p for all p < l/(l 1). (Note that this regularity is the best possible since the density is never L l/(l 1).) Much more recently, Bruin, Shen and myself proved that no growth condition is needed at all: 24

25 Theorem 10. In fact there exists constant C(f) such that if lim inf n (f n ) (f(c)) C for each critical point c then f has an acip. 4.1 Strategy of proof of the theorem on the existence of acim s without growth conditions We say that f satisfies the backward contracting property with constant r (BC(r) in short) if the following holds: there exists ɛ 0 > 0 such that for each ɛ < ɛ 0, any critical points c, c Crit(f) and any component W of f s (B rɛ (f(c ))), s 1 W B ɛ (f(c)) implies W ɛ. (1) We say that f satisfies BC( ) if it satisfies BC(r) for all r > 1. Clearly, for any r > 1 the property BC(r) implies that f has no critical relation, i.e., no critical point is mapped into the critical set under forward iteration. 25

26 The proof of Theorem 10 breaks into the following two propositions. Proposition 1. If lim inf n (f n ) (f(c)) C for each critical point c then f satisfies property BC(r) where r depends on C. Proposition 2. There exists r(f) such that if f satisfies the BC(r) then there exists a constant M such that for every Borel set A we have f n (A) M fa κ/l max. (2) It follows from the inequality (2) that any weak limit µ of 1 n n 1 (f i ) Leb i=0 is an acip of f. In fact the density of µ with respect to the Lebesgue measure is in L p, where p l max /(l max 1) as κ 1. So the previous propositions imply the theorem. 26

28 Note: Negative Schwarzian Sf = f /f (3/2)(f /f ) 2 < 0 looks a strong assumption. It is NOT. One can prove that any map which has no neutral periodic orbits is smoothly conjugate to one which has negative Schwarzian (modulo some mild other smoothness assumptions.) 28

29 4.2 The proof of Proposition 2 The proof Proposition 2 goes in several steps: Step 1: An open set V [0, 1] is called nice if for each x V and for any k 1, f k (x) V. For λ > 0 we say that a nice open set V is λ-nice if for each return domain J of V, we have (1 + 2λ)J V. Lemma 1. For any f A and each λ > 1 there exists r > 2 such that if f satisfying BC(r) and any ɛ > 0 sufficiently small, the following holds: for each c Crit(f), there exists an open interval V c such that V := c Crit(f) V c is nice and such that for each c Crit(f) we have ˆB ɛ (c) V c ˆB 2ɛ (c); V is λ-nice. Here ˆB ɛ (c) is defined to be the connected component of f 1 (B ɛ (f(c))) containing c. 29

30 Proof. The proof follows from the following argument due to Rivera- Letelier. For ɛ > 0 small, define the open set V n = n i=0 f i ( ˆB ɛ ). Clearly V is nice. Take Vc n to be the connected component of V n which contains c, and let V c = Vc. It remains to show that Vc n ˆB 2ɛ (c) for each n. We do this by induction. For n = 0 this holds by definition, so assume it holds for n. Consider Z = f(vc n+1 ) \ B(f(c), ɛ). For z Z, there exists m(z) {0, 1,..., n} and c 0 (z) Crit(f) so that f m(z) (z) ˆB ɛ (c 0 (z)). Now choose z 0 Z so that m 0 = m(z 0 ) is minimal among m(z) for points z Z and let ĉ 0 = c 0 (z). Since f m 0(z) ˆB ɛ (ĉ 0 ), and since f m 0(Z) Vc n, the induction hypothesis implies f m 0 (Z) ˆB 2ɛ (ĉ 0 ). Since f satisfies BC(2) and Z has distance ɛ to f(c), it follows that f(z) < ɛ and Z ˆB 2ɛ (c). This completes the induction step. To prove that V is λ-nice one uses some real bounds. The following notion will then be useful. 30

31 If I is an interval which contains a critical point c and J is a uni-critical pull back of I then we say that J is a child of I. So there exists c Crit(f), s 0 and an interval J f(c ) such that f s maps J diffeomorphically onto I; J is the component of f 1 ( J) which contains c. Lemma 2. Let c, c Crit(f), let I c be a λ-nice interval and let J 1 J 2 J m be children of I which contain c. Then f(j i ) ρ i 1 f(j 1 ) holds for all i, where ρ = ρ(λ) > 0 is a constant, and ρ 0 as λ. Step 2: Assume that f satisfied BC(r). Then by Lemma 1 there exists a λ-nice puzzle piece I between ˆB ɛ and ˆB 2ɛ. By Lemma 2 we know that the children of this set I shrink fast. 31

32 Set r 1 = r 1/l max. For each n 0 and δ > 0, define L n (δ) = sup{ f m (A) : 0 m n, A ˆB ɛ0 /r 1 is an interval, fa δ}. Lemma 3. Let I be a λ-nice interval such that ˆBɛ (c) I ˆB 2ɛ (c), where c Crit(f) and ɛ < ɛ 0, and let A be an interval such that Then for all n 1, A ˆB ɛ/2 (c) and A ˆB ɛ/r1 (c). (3) f n (A) C A I f n (I) + 2N L n 1 (ρ i fa ), (4) The proof goes by splitting collection of all components J of f n (I) into the class J 0 of components J of f n (I) such that f n : J I is a diffeomorphism and J 1 of all other components of f n (I). By the Koebe principle, for all J J 0, we have f n (A) J J i=1 where C is a constant depending only on l max. 32 C A I, (5)

33 For each J J 1, there exist a unique 0 n 1 = n 1 (J) < n and a unique interval J f n n 1(J) such that J is a child of I and such that f n n 1(J) contains a critical point. Let A a component of f n 1(A) J then by Koebe f(a ) f(j ) C A f n 1 (J ) C A I C (2r 1 ) l c 1 f(a) f(i) where the 2nd inequality holds because f n 1(J ) contains a component of I \ A and last inequality because of non-flatness of f and (3). Because J is a child and using Lemma 2 the claim follows. Step 3: Using some induction argument it then follows that for each interval A inside some small neighbourhood U of Crit(f), f n (A) K A κ/l max. To get this estimate for general sets one uses a combinatorial argument and the minimum principle: (the derivative is closest to the boundary of a branch). 33

34 5 Ubiquity of having a physical measure In the one-dimensional setting the situation is better understood. Theorem 11 (Existence of SRB measures). For Lebesgue almost all parameters c R, the map R z z l + c has a unique SRB measure, which is either absolutely continuous or its support is equal to ω(0) (the omega-limit set of the critical point 0) and f ω(0) is uniquely ergodic. This was proved by Lyubich for the case that l = 2, and for l any even integer this was proved by Bruin, Shen and SvS. In fact for l = 2, for typical parameters c R the SRB measure is absolutely continuous or concentrated on an attracting hyperbolic periodic orbit. 34

35 Our proof of the previous theorem works also for general onedimensional maps, except in one place. Hopefully this is just a technical difficulty Strategy of proof We do not aim to get positive Lyapounov exponents. To get an acim, we are showing that for all parameters c f c has low combinatorial complexity and (therefore) a unique physical measure, or f c has the decay of geometry condition. Then we show that for almost all parameters c which have the decay of geometry condition the following summability condition SC holds: 1 < for any α > 0. (6) Dfc n (c) α n=0 As we have seen such parameters have an absolutely continuous invariant probability measure. 35

36 5.2 Reluctant versus persistent recurrence Let us first define some parameter sets. As mentioned, an interval J 0 is a called a child of I if it is a unimodal pullback of I, i.e., if there exists an interval J containing the critical value c and an integer s 0 so that f s 1 : J I is a homeomorphism and J = f 1 ( J). If f c is non-renormalizable, recurrent and there exists a nice interval I 0 with infinitely many children, then we say that f c is reluctantly recurrent; otherwise it is called persistently recurrent. Reluctantly recurrent means that you can go from arbitrarily small neighbourhoods of the critical value diffeomorphically to big scale. This is the easy case. 36

37 5.3 The Decaying Geometry Condition Let us say that a parameter c for which the critical point of f c is recurrent and f c is non-renormalizable, has decaying geometry property if either f c is reluctantly recurrent, or f c is persistently recurrent and there exists a sequence of nice intervals Γ 0 Γ 1 0 such that for each n 0, Γ n+1 is the smallest child of Γ n, and so that Γ n+1 / Γ n 0 as n. Let DG denote the collection of parameters c for which f c satisfies the decaying geometry condition. We should note that if l = 2, all parameters have the decaying geometry property. This is definitely NOT true when l > 2. 37

38 5.4 Combinatorial classification Without going into details, we define the notion of low combinatorial complexity and show Theorem 12. If f is persistently recurrent and has low combinatorial complexity, then f ω(0) is uniquely ergodic. Moreover, if f is C 3, then there is a unique physical-measure which is either an acip, or is the unique invariant probability measure supported on ω(0). It follows from Lebesgue ergodicity of unimodal maps without periodic attractors that an acip is indeed the unique physical-measure. The link with the decaying geometry property is made in the following: Theorem 13. If c / DG, then f c has low combinatorial complexity (and hence Theorem 12 applies). So it is enough to consider only parameters in DG. 38

39 5.5 Parameters in DG Pick a parameter c DG. Then by definition either the critical point is reluctantly recurrent (and one can go to big scale) or it is persistently recurrent and there exists a sequence of nice intervals Γ 0 Γ 1 0 such that for each n 0, Γ n+1 is the smallest child of Γ n, and so that Γ n+1 / Γ n 0 as n. From this one can show that there exists a critical puzzle piece V c so that the first return map g c to V c g c : i U i,c V c has the property that the moduli of V c \ U i,c are as large as we want. 39

40 To state a sharper version of this, we first give some definitions. Given a topological disk Ω C and a set A, define λ(a Ω) = sup ϕ m(φ(a Ω)), m(φ(ω)) where ϕ runs over all conformal maps from Ω into C and m denotes the planar Lebesgue measure. Definition 1. Let V C be a topological disk. Let U i, i = 0, 1,... be pairwise disjoint topological disks contained in V. We say that the family {U i } is ɛ-absolutely-small in V if λ( i U i V ) < ɛ, and for each i, the diameter of U i in the hyperbolic metric of V is less than ɛ. Here the hyperbolic metric of V is the pullback of the standard Poincaré metric on the unit disc D C by the Riemann mapping from V to D. Theorem 14. Consider a map f = f c with c DG. Then for any ɛ > 0, there exists a critical puzzle piece Y such that the collection of the components of the domain of the first return map to Y is ɛ-absolutely-small in Y. 40

41 5.6 Robustness under changes in the parameter Now consider c 0 as above and the first return map g c0 : i U i,c0 V c0 := Y Also consider the set of parameters D c 0 so that the puzzle piece V c (consisting of rays and equipotentials) still exists (moves holomorphically) for all c D and so that the first return g c : i Ũ i,c Ṽc is pseudo-conjuate to g c0 : i U i,c 0 V c0. Here a qc map φ : C C is called a pseudo-conjugacy between these maps if φ maps V onto Ṽ, U i onto Ũi, and respects the boundary dynamics: for each z U i, φ g(z) = g φ(z). 41

42 Proposition 15. Let g and g be first return maps as above, and let φ be a pseudo-conjugacy between them which is conformal a.e. in V \ ( U i ). There exists a universal constant ɛ 0 > 0 such that provided that {U i } is ɛ 0 -absolutely-small in V, there exist a 2-qc map χ 0 : V Ṽ such that χ 0 = φ on V ; a qc pseudo-conjugacy map ψ : C C such that ψ = φ on C \ i U i such that ψ is 2-qc on V \ U 0. So the information on g c0 : i U i,c0 V c0 := Y holds also for g c : i U i,c V c for all c D. As c D goes to the boundary the domain U 0,c containing the critical point degenerates. That is why ψ is 2-qc only on V \ U 0. 42

43 5.7 Decay of Geometry and the SC condition To get that the SC condition holds for almost all parameters in DG, we use the following lemma: Lemma 4. Let g : U i V be a polynomial-like map such that U 0 contains the critical point of g. Let W be a domain of the first return map to U 0 (under g) with return time s. Then mod(u 0 \ W ) 1 l ((s 1)mod (g) + mod(g)) (7) where we define mod(g) = mod(v \U 0 ) and mod (g) = inf i mod(v \ U i ). So provided the first return map visits sufficiently many noncentral domains, the modulus grows. So to get maps which most maps are in SC it is enough to show that maps so that the critical point only visits a bounded number of central domains at each step, are rare. 43

45 5.9 Parameter removal We start with a map g c in DG. From this it follows that if we consider the first return map g c, c D to appropriate critical puzzle pieces P n (c) g = {g c i U i,c V c, c D} then the modulus (in the sense as before) is as large as we want. It is easy to show that there is a holomorphic motion of these puzzle pieces and that c g c (0) forms a diagonal. Then we apply an inductive scheme where we remove the parameters which only visit a small number of domains. the polynomial-like parameters which we keep, will have exponentially growing moduli. This implies that SC holds. since the moduli grow, the proportion of the parameters which we remove also gets exponentially smaller. 45

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